Seasonal ice-cream sales and a badly made forecast
An ice-cream kiosk records its quarterly sales, in hundreds of ice creams, over four years.
Year 1: 17, 35, 52, 28 Year 2: 25, 43, 60, 36 Year 3: 33, 51, 68, 44 Year 4: 41, 59, 76, 52
The new manager says: 'In Year 4 we sold 41 in the first quarter and 76 in the third - demand nearly doubled in six months and is out of control. I will plan for about 76 in the first quarter of Year 5.'
1. Explain why the manager's comparison of Q1 and Q3 of Year 4 is misleading, and use the 16 values to describe the real trend and seasonal pattern.
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Why it is misleading: the manager is comparing two different seasons, so the change measured is mostly seasonal variation, not growth. In every year Q3 is the highest quarter and Q1 the lowest, and the Q3 − Q1 gap is identical each year: 52 − 17 = 35, 60 − 25 = 35, 68 − 33 = 35, 76 − 41 = 35. That constant 35 is the seasonal swing.
The real trend is found by comparing the same quarter in successive years:
- Q1: 17, 25, 33, 41 - rising by 8 each year
- Q2: 35, 43, 51, 59 - rising by 8 each year
- Q3: 52, 60, 68, 76 - rising by 8 each year
- Q4: 28, 36, 44, 52 - rising by 8 each year
So there is a steady upward trend of about 8 hundred (800 ice creams) per year, about 2 per quarter, with a fixed seasonal pattern on top.
Seasonal effects: quarter means are Q1 29, Q2 47, Q3 64, Q4 40, with overall mean 45, giving average seasonal variation of about −16, +2, +19 and −5 (these sum to zero). Q3 sits about 19 above trend; Q1 about 16 below.
2. Produce forecasts for Q1 and Q3 of Year 5, and explain why a single line of best fit drawn through all 16 points should not be used to forecast a particular quarter.
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Forecasts (using the same quarter one year earlier, which removes the seasonality):
- Year 5 Q1 = 41 + 8 = 49 hundred ice creams
- Year 5 Q3 = 76 + 8 = 84 hundred ice creams
(Equivalently, deseasonalise the last Q1: 41 + 16 = 57 on trend, add 8 of trend to get 65, then take off the −16 seasonal effect: 65 − 16 = 49.)
Why not one line through all 16 points: such a line describes the average level of a quarter, not any particular quarter. Its predictions mix the trend with the seasonal swing, so a prediction for Q1 of Year 5 would be pulled up towards the level of the middle quarters and would over-predict Q1 by roughly 16 hundred - the size of the Q1 seasonal effect. To forecast one named quarter you must first remove the seasonal variation, forecast the trend, then add the seasonal effect back.
Assumption to state: the trend continues to rise by about 8 per year and the seasonal pattern stays the same - neither is guaranteed (a heatwave, a new competitor or a price rise would break the model).